Notice that for hyperbolic rational homology spheres \(Y\) , a lower bound on \(\lambda _1^*(Y)\)  implies a lower bound on \(\mathrm {inj(Y)}\)  which can be explicitly computed (see [1]}). In particular, an analogue of Theorem REF  can be stated for the subgroup \(\Theta _{V,\delta }\)  generated by hyperbolic homology spheres \(Y\) with \(\mathrm {vol}(Y)\le V\)  and \(\lambda _1^*(Y)\ge \delta \) . We decided to keep the dependence on \(\mathrm {inj(Y)}\)  in the statement of Theorem REF  because the injectivity radius is a very natural quantity to study and plays an explicit role in several points in our argument.
